Regular and Irregular Multi-resolution Terrain Models: a Comparison

1. Regular and Irregular Multi-resolution Terrain Models: a Comparison Leila De Floriani * Paola Magillo Department of Computer Science University of Genova, Genova (Italy) * currently at the University of Maryland, College Park, MD

2. Outline • Motivations • Related Work • Regular and Irregular Multi-Triangulations (MTs) • Vertex-Based Multi-Triangulation (Vertex-based MT) • Hierarchy of Right Triangles (HRT) • Level-Of-Detail (LOD) Queries for Terrain Modeling • Data Structures • Experimental Results and Comparisons • Summary and Future Work

3. Why Multi-resolution Terrain Models? • Large-size terrain data sets • high storage requirements and high processing times • Multi-resolution terrain models • encompass a collection of terrain representations at different levels of resolutions

4. Why Multi-resolution Terrain Models? • Multi-resolution terrain models • allow extracting terrain representations at a variable resolution

5. Regular and Irregular Multi-Resolution Models • Data on a grid versus scattered data • Regular versus irregular multi-resolution models: • Regular models: based on a nested domain decomposition • Irregular models: compact way of encoding the steps performed by a simplification process applied to an irregular triangle mesh

6. Regular and Irregular Multi-Resolution Models • Both are instances of a common framework: the Multi-Triangulation • Our objective: compare and analyze regular and irregular multi-resolution models • Comparison: • size of the models • space requirements of their encoding structures • efficiency in extracting meshes at variable resolution: Level-Of-Detail (LOD) queries

7. Related Work • Regular multi-resolution models: • Triangle quadtrees (Gomez and Guzman, 1979; Dutton, 1983) • Restricted quadtrees (Von Herzen and Barr, 1987; Sivan and Samet, 1992) • Hierarchies of right triangles (Duchaineau et a., 1997; Evans et al., 2001; Lindstrom et al., 1996; Pajarola, 1998) • Irregular multi-resolution models: • Nested meshes (De Floriani and Puppo, 1995; Scarlatos and Pavlidis, 1994) • Pyramidal triangle meshes (De Berg and Dobrindt, 1995; De Floriani, 1989) • Progressive meshes (Hoppe, 1996; Taubin et al., 1998) • Continuous LOD models (Hoppe, 1998; Xia et al., 1997; Maheshvari et al.,1997; El Sana and Varshney 1999: De Floriani et al., 1998)

8. Basic Concepts: Modifications • Modification of a triangle mesh: replace a connected set of triangles with other triangles covering the same region

9. Basic Concepts: Dependencies • A modification M2 depends on a modification M1 iff M2 changes some triangles that have been changed by M1 M2 depends on M1 M2 does not depend on M1

10. The Multi-Triangulation (MT) • A base mesh • A set of modifications • A partial order (dependency relation)

11. The Multi-Triangulation (MT) • Vertex-Based Multi-Triangulation (Vertex-based MT): • Scattered data • Modification: vertex insertion • Hierarchy of Right Triangles (HRT): • Data on a grid • Modification: simultaneous bisection of two adjacent right triangles

12. LOD Queries • A set of basic queries for analysis and visualization of a terrain at different levels of detail • Instances of selective refinement: extract from a Multi-triangulation a mesh with the smallest possible number of triangles satisfying some user-defined criterion based on LOD • LOD based on approximation error • LOD can be uniform on the whole domain,or variable at each point of the domain.

13. Data Structures for Multi-Triangulations • They must support efficiently: • do/undo modifications on the extracted mesh • test dependency links (to decide whether a modification can be applied) • Data Structures for Vertex-Based MTs: • Procedural encoding of modifications • Partial order represented as a Directed Acyclic Graph (DAG) • Approximation error associated either with triangles or with modifications

14. Procedural Encoding of Modifications in a Vertex-based MT • To insert a vertex: • Recognize the triangles to be deleted – hard • Create triangles incident in the new vertex – easy • To remove a vertex: • Delete triangles incident in the vertex – easy • Reconstruct the triangles inside the hole – hard

15. Encoding a Triangulated Polygon • Encode an anchor edge • Perform a depth-first traversal as a tree • Encode the traversal as a bit stream 10 00 11 11

16. Building a Vertex-based MT • A vertex-based MT is built through error-driven techniques based on • coarsening an initial mesh through vertex insertion (VI) • decimating the full-resolution mesh through vertex removal • iterative removal of a single vertex (VR) • removal of a set of independent vertices (IVR): • Shape of a vertex-based MT (number of triangles, size of modifications) depends on its construction strategy • Size of the modifications with different strategies: • MT-VI : each modification creates 5 triangles on average • MT-IVR: each modification creates 5.5 triangles on average • MT-VR : each modification creates 6 triangles on average

17. Encoding a Hierarchy of Right Triangles • Modifications and dependency links are implicitly represented • Each triangles is uniquely identified by a binary location code • From the location code of a triangle t we can retrieve: • vertex coordinates and height values • modifications involving t • dependency links for such modifications • Only height values and errors (associated with triangles) are stored

18. Comparison:Size of an MT (number of triangles)

19. Comparison:Storage Costs of the Data Structures • Full-resolution mesh (encoded in a standard triangle- based data structure) : 54n bytes • Vertex-based MT: 27n (error on modifications) 33n bytes (error on triangles) • between 1/2 and 3/5 of the space required by mesh at full resolution • Hierarchy of right triangles:6n bytes • 1/5 of the space required by a vertex-based MT • 1/9 of the space required by mesh at full resolution

20. Comparison:Level-Of-Detail (LOD) Queries • Uniform LOD across the domain • Variable LOD: • Domain-based LOD: max resolution inside a window • Field-based LOD: max resolution for selected contour values • Best solution = fewer extracted triangles for the same error value

21. Comparison: Uniform LOD • MT-VI is the best one • MT-VR, MT- IVR, HRT are comparable • Motivation: error-driven construction strategy HRT - VI - IVR - VR - HRT - VI - IVR - VR - Mount Marcy Devil Peak

22. Comparison: Uniform LOD error = 1.3% of height range HRT 22045 triangles MT-VI 16208 triangles

23. Comparison: Uniform LOD error = 6.7% of height range HRT 3648 triangles MT-VI 1951 triangles

24. Comparison: Variable LOD • HRT is the best one • MT-VI, MT-IVR are comparable (MT-IVR slightly better) • MT-VR is the worst one • Motivation: smaller modifications, fewer dependency links (HRT = each modification creates 4 triangles) HRT - VI - IVR - VR - HRT - VI - IVR - VR - Window focus Mount Marcy Devil Peak

25. Comparison: Variable LOD • HRT gives the best results • MT-VI, MT-IVR are comparable (MT-VI slightly better) • MT-VR give the worst results • Motivation: smaller modifications, fewer dependency links (HRT = each modification creates 4 triangles) HRT - VI - IVR - VR - HRT - VI - IVR - VR - Field focus Mount Marcy Devil Peak

26. Comparison: Variable LOD error = 1.3% of height range, focused inside a window HRT 1614 triangles MT-VI 2072 triangles

27. Comparison: Variable LOD error = 1.3% of height range, focused on a field value HRT 6697 triangles MT-VI 7138 triangles

28. Summary • Both can generate meshes with topology (triangle-triangle adjacency links)

29. Current and Future Work • Out-of-core techniques for a vertex-based MT: data structures, simplification methods, query algorithms • 3D extension for volume data visualization: • Efficient neighbor-finding techniques for a hierarchy of tetrahedra (Lee, De Floriani and Samet, 2001) • Compact data structures for irregular 3D MTs (De Floriani et al., 2002) • Analysis and comparison of regular and irregular 3D MTs on large volume data sets

30. Generating a Vertex-based MT • Error-driven vertex insertion (VI)top-down • At each step insert the data point corresponding to the maximum error (maximum vertical distance from the existing surface) • Error-driven vertex removal (VR) bottom-up • Start with a full-resolution mesh • At each step remove the vertex whose removal causes the least error increase (minimum vertical distance from the new surface) • Error-driven independent vertex removal (IVR) bottom-up • Start with a full-resolution mesh • At each step remove an independent set of vertices selected as the ones causing the least error increase

31. Generating a Vertex-based MT • Error-driven vertex insertion (VI)top-down • At each step insert the data point corresponding to the maximum error (maximum vertical distance from the existing surface)

32. Generating a Vertex-based MT • Error-driven vertex removal (VR) bottom-up • Start with a full-resolution mesh • At each step remove the vertex whose removal causes the least error increase (minimum vertical distance from the new surface)

33. Generating a Vertex-based MT • Error-driven independent vertex removal (IVR) bottom-up • Start with a full-resolution mesh • At each step remove an independent set of vertices selected as the ones causing the least error increase

34. Space Requirements for a Vertex-based MT • Storage cost: • 27n bytes (errors associated with modifications) • 33n bytes (errors associated with triangles) where n = number of vertices in the data set by assuming (pessimistic experimental estimates) • number of triangles = 4n • number of arcs in the DAG = 3n • Size (number of triangles) depends on the error-driven MT construction strategy: • vertex insertion (VI) • vertex removal (VR) • removal of a set of independent vertices (IVR)